022. Equivalent Fractions and Simplifying Them

Learning Intentions

• To understand what it means for two fractions to be equivalent
• simplify fractions by dividing by their highest common factor

Pre-requisite Summary

• Understand that a fraction represents equal parts of a whole
• Know the meaning of numerator and denominator
• Be able to identify factors of whole numbers
• Understand that multiplying or dividing two numbers by the same non-zero whole number can preserve a relationship
• Be able to find the highest common factor of two numbers
• Recognise that fractions can name the same amount in different ways

Worked Examples

Worked Example 1

a) Explain what it means for ${\displaystyle \frac{1}{2}}$ and ${\displaystyle \frac{2}{4}}$ to be equivalent.
b) Decide whether ${\displaystyle \frac{3}{5}}$ and ${\displaystyle \frac{6}{10}}$ are equivalent.
c) Justify your answer.

Worked Example 2

a) Find two fractions equivalent to ${\displaystyle \frac{2}{3}}$.
b) Find two fractions equivalent to ${\displaystyle \frac{3}{4}}$.

Worked Example 3

Simplify each fraction by dividing by the highest common factor:
a) ${\displaystyle \frac{8}{12}}$
b) ${\displaystyle \frac{15}{20}}$

Worked Example 4

Simplify each fraction by dividing by the highest common factor:
a) ${\displaystyle \frac{18}{27}}$
b) ${\displaystyle \frac{24}{36}}$

Worked Example 5

a) Decide whether ${\displaystyle \frac{4}{6}}$ and ${\displaystyle \frac{6}{9}}$ are equivalent.
b) Simplify both fractions.
c) Explain how the simplified forms help.

Worked Example 6

a) Simplify ${\displaystyle \frac{28}{42}}$ by dividing by the highest common factor.
b) Explain why the simplified fraction is equivalent to the original fraction.

Problems

Problem 1

a) Explain what it means for ${\displaystyle \frac{1}{3}}$ and ${\displaystyle \frac{2}{6}}$ to be equivalent.
b) Decide whether ${\displaystyle \frac{2}{7}}$ and ${\displaystyle \frac{4}{14}}$ are equivalent.
c) Justify your answer.

Problem 2

a) Find two fractions equivalent to ${\displaystyle \frac{3}{5}}$.
b) Find two fractions equivalent to ${\displaystyle \frac{4}{7}}$.

Problem 3

Simplify each fraction by dividing by the highest common factor:
a) ${\displaystyle \frac{10}{15}}$
b) ${\displaystyle \frac{14}{21}}$

Problem 4

Simplify each fraction by dividing by the highest common factor:
a) ${\displaystyle \frac{16}{24}}$
b) ${\displaystyle \frac{21}{35}}$

Problem 5

a) Decide whether ${\displaystyle \frac{6}{8}}$ and ${\displaystyle \frac{9}{12}}$ are equivalent.
b) Simplify both fractions.
c) Explain how the simplified forms help.

Problem 6

a) Simplify ${\displaystyle \frac{30}{45}}$ by dividing by the highest common factor.
b) Explain why the simplified fraction is equivalent to the original fraction.

Exercises

Understanding and Fluency

1. State whether each pair of fractions is equivalent:
   a) ${\displaystyle \frac{1}{2}}$ and ${\displaystyle \frac{2}{4}}$
   b) ${\displaystyle \frac{2}{3}}$ and ${\displaystyle \frac{4}{6}}$
   c) ${\displaystyle \frac{3}{4}}$ and ${\displaystyle \frac{6}{8}}$

2. State whether each pair of fractions is equivalent:
   a) ${\displaystyle \frac{2}{5}}$ and ${\displaystyle \frac{4}{10}}$
   b) ${\displaystyle \frac{3}{7}}$ and ${\displaystyle \frac{6}{15}}$
   c) ${\displaystyle \frac{5}{6}}$ and ${\displaystyle \frac{10}{12}}$

3. Find two fractions equivalent to each fraction:
   a) ${\displaystyle \frac{1}{4}}$
   b) ${\displaystyle \frac{2}{5}}$
   c) ${\displaystyle \frac{3}{8}}$

4. Find two fractions equivalent to each fraction:
   a) ${\displaystyle \frac{4}{9}}$
   b) ${\displaystyle \frac{5}{7}}$
   c) ${\displaystyle \frac{3}{10}}$

5. Simplify each fraction by dividing by the highest common factor:
   a) ${\displaystyle \frac{6}{9}}$
   b) ${\displaystyle \frac{12}{18}}$
   c) ${\displaystyle \frac{20}{30}}$

6. Simplify each fraction by dividing by the highest common factor:
   a) ${\displaystyle \frac{14}{35}}$
   b) ${\displaystyle \frac{18}{24}}$
   c) ${\displaystyle \frac{27}{36}}$

7. Simplify each fraction by dividing by the highest common factor:
   a) ${\displaystyle \frac{16}{20}}$
   b) ${\displaystyle \frac{24}{32}}$
   c) ${\displaystyle \frac{45}{60}}$

8. Decide whether the fractions are equivalent by simplifying:
   a) ${\displaystyle \frac{4}{10}}$ and ${\displaystyle \frac{6}{15}}$
   b) ${\displaystyle \frac{8}{12}}$ and ${\displaystyle \frac{10}{15}}$
   c) ${\displaystyle \frac{9}{14}}$ and ${\displaystyle \frac{18}{28}}$

Reasoning

9. Explain why multiplying the numerator and denominator of a fraction by the same whole number gives an equivalent fraction.

10. A student says ${\displaystyle \frac{2}{3}}$ and ${\displaystyle \frac{2}{6}}$ are equivalent because both numerators are $2$. Explain the mistake.

11. Explain why dividing the numerator and denominator by the highest common factor gives the fraction in simplest form.

12. A student simplifies ${\displaystyle \frac{12}{18}}$ to ${\displaystyle \frac{6}{18}}$. Explain why this is not fully simplified.

Problem-solving

13. A pizza is cut into $8$ equal slices and $4$ are eaten. Write the fraction eaten and then simplify it.

14. A ribbon is divided into $12$ equal parts and $8$ parts are coloured. Write the fraction coloured and simplify it.

15. A class completed ${\displaystyle \frac{15}{20}}$ of a task. Write this fraction in simplest form.

16. On a number line, one point is labelled ${\displaystyle \frac{3}{4}}$ and another is labelled ${\displaystyle \frac{6}{8}}$. Explain why they are at the same location.

17. A rectangle has $18$ equal regions and $12$ are shaded. Write the shaded fraction and simplify it.

18. A fraction is equivalent to ${\displaystyle \frac{2}{5}}$ and has denominator $25$. Find the fraction and explain how you know.

Potential Misunderstandings

• Students may think equivalent fractions must have the same numerator or the same denominator
• Students may change only the numerator or only the denominator when generating equivalent fractions
• Students may not recognise that equivalent fractions represent the same quantity
• Students may simplify by subtracting rather than dividing
• Students may divide the numerator and denominator by different numbers
• Students may stop simplifying before the fraction is in simplest form
• Students may not use the highest common factor and therefore may simplify only part of the way
• Students may confuse simplifying a fraction with changing its value

Next: 023. Improper Fractions and Mixed Numerals